Method and apparatus to determine golf ball trajectory and flight

ABSTRACT

A launch monitor system including a support structure, a first light-reflecting element disposed on this support structure, a lighting unit and an camera unit. A computer receives signals generated by light patterns received by the camera unit and computes a variety of flight characteristics for the object. The system may be moved back and forth to vary the field-of-view of the camera unit. The system also computes and displays object trajectories from the computed flight characteristics which account for the characteristics of the object and the atmospheric conditions.

[0001] This application is a continuation application of U.S.application Ser. No. 09/566,803 filed on May 8, 2000, now pending, whichis a divisional application of U.S. application Ser. No. 09/156,611filed on Sep. 18, 1998, now U.S. Pat. No. 6,241,622, both of which areincorporated herein by reference in their entirety.

BACKGROUND OF THE INVENTION

[0002] Apparatus for measuring golf ball flight characteristics areknown (U.S. Pat Nos. 4,063,259; 4,375,887; 4,158,853; and 4,136,387).Techniques of detecting golf club head position and golf ball positionshortly after impact using photoelectric means to trigger a flash topermit a photograph to be taken of the club head have been disclosed(U.S. Pat. Nos. 4,063,259 and 4,375,887). Golf ball or golf club headmovement has been determined by placing reflective areas on a golf ballalong with use of electro-optical sensors (U.S. Pat. No. 4,136,387). Theelectro-optical sensing of light sources on both the golfer's body andclub has also been disclosed (U.S. Pat. No. 4,137,566). In addition,apparatus for monitoring a golfer and the golf club being swung has alsobeen disclosed (U.S. Pat. No. 4,137,566).

[0003] One particularly troublesome aspect of past systems for measuringgolf ball flight characteristics relates to their lack of portability.In this regard, prior systems have generally required cameras, sensorsand strobe lights set up in various positions about the golfer. Inaddition, past systems have not had the ability to be utilized outdoorsbut have had to be set up indoors under less than ideal or realisticgolfing conditions. As prior golf ball and/or golf club monitoringsystems have not been portable and have not been capable of practicaluse outdoors, the systems have not been usable in the most desirableteaching or club fitting locations, e.g., on an outdoor driving range.Also, while the systems disclosed in the related applications andpatents mentioned above, which are incorporated herein by reference) areportable and are capable of use outdoors, further improvements relatedto increased portability would be desirable to allow easiertransportation of the unit between sites and easier movement of the unitat any particular site.

[0004] One additional area that has not been adequately addressed bypast golf ball launch monitoring systems relates to the area ofpredicting flight path differences based on different physicalcharacteristics of golf balls and/or different atmospheric conditionsthat a golfer may encounter after being tested by the launch monitorsystem. It would therefore be desirable to provide a system whichmeasures the launch or flight characteristics of a golf ball having aparticular construction, such as a two-piece construction and underideal atmospheric conditions and then provide the golfer with revisedgolf ball flight results based on computer predictions for golf ballshaving different physical characteristics (such as a three-piece golfball) and different atmospheric conditions (such as higher elevations,higher humidity or more adverse wind conditions).

SUMMARY OF THE INVENTION

[0005] Broadly, the present invention comprises method and apparatus formeasuring the speed, direction, and orientation of a golf ball and fromsuch data computing the flight path of the golf ball.

[0006] It is a feature that the method and apparatus particularly applyto golf equipment and that the present invention provides a golfer withdata relating to the variables of his swing useful in improving theswing and in selecting advantageous equipment for use, including thetypes of golf balls.

[0007] In particular, the present invention contemplates a launchmonitor system for measuring launch characteristics of a golf ball fromdata taken when the golf ball is in a predetermined field-of-view. Thesystem preferably includes a support structure, which is a single,portable support structure, light-reflecting elements disposed on thesupport structure, a lighting unit, and camera units. The lighting unitincludes a light source directed at the light-reflecting elements forreflecting light into the predetermined field-of-view, and theelectro-optical units, disposed on the support structure in proximity tothe light-reflecting elements, are directed toward the predeterminedfield-of-view. The light-reflecting elements include an aperture and thecamera units are disposed to monitor the predetermined field-of-viewthrough the aperture.

[0008] This allows, for example, the system to be used outdoors in agrassy area and for the hitting area to be slightly varied to move theplayer away from divots, etc. The support elements may include slidepads, wheels, or combinations of both. The support elements may beheight-adjustable to vary the orientation and direction of view of thesystem and, specifically, the camera units. As an additional aspect ofthe invention, a distance calibrator is provided for calibrating thedistance between the camera unit or units and the predeterminedfield-of-view.

[0009] It is an object of the system to measure launch characteristicsof an object after it is struck by a striking instrument.

[0010] It is a further object of the invention to provide computingmeans to calculate the trajectory of the object from the launchcharacteristics and information on the environmental conditions and theobject's characteristics and to generate statistics on these objecttrajectories.

[0011] It is a further object of the invention to provide a controlmeans to manage the tasks performed by the system including cameraactivation, shutter control, image capture, calculation of launchcharacteristics, calculation of object trajectories, and generation ofobject trajectory statistics.

BRIEF DESCRIPTION OF THE DRAWINGS

[0012]FIG. 1 is a perspective view of a first embodiment of the presentinvention;

[0013]FIG. 2 is a top view thereof;

[0014]FIG. 3 is a side elevational view of the system shown in FIGS. 1and 2;

[0015]FIG. 4 is an elevational view of the light receiving and sensorygrid panel located in each camera;

[0016]FIG. 5 is a perspective view of a three-dimensional rectilinearfield showing a golf ball at two different positions I and II;

[0017]FIG. 6 is a perspective view of a second embodiment of thisinvention;

[0018]FIG. 7 is a top view of the apparatus shown in FIG. 6 andgenerally showing calibration of the system;

[0019]FIG. 8 is a side elevational view of the system shown in FIGS. 6and 7;

[0020]FIG. 9 is a top view of the system shown in FIGS. 6-8 andgenerally showing a golf ball in place under operating conditions;

[0021]FIG. 10 is a partial, cut-away top view of the system shown inFIGS. 6-9 illustrating the strobe lighting unit;

[0022]FIGS. 11A and 11B is a diagram of a voltage amplifier andregulator circuit and a trigger and discharge circuit, respectively,used in the system shown in FIGS. 6-10;

[0023]FIG. 12 is an example of a Fresnel lens used in the presentinvention;

[0024]FIG. 13 illustrates a light pattern without a Fresnel lens;

[0025]FIG. 14 illustrates a light pattern with a Fresnel lens;

[0026]FIG. 15 is a perspective view of the calibration fixture carryingfifteen illuminable areas;

[0027]FIG. 16 is a perspective view of an unassembled rod useful forallowing movement of a system constructed in accordance with theinvention;

[0028]FIG. 17 is an elevational view of the rod of FIG. 16 shown in anassembled condition;

[0029]FIG. 18 is a flow chart describing the operation of the system;

[0030]FIG. 19 is a flow chart describing the calibration of the system;

[0031]FIG. 20 is a flow chart describing the determination of dots inthe image;

[0032]FIG. 21 is a graph showing the trajectory of the golf ball ascalculated by the system; and

[0033]FIG. 22 is an example of a contour map of the total distance agolf ball travels under specified conditions.

DESCRIPTION OF THE PREFERRED EMBODIMENT

[0034]FIG. 1 illustrates a preferred first embodiment of the inventionin the form of a portable launch monitoring system 10 including a baseor support structure 12 and attached support elements 14, 16. Supportelements 14, 16 are specifically shown as slide pads each includingV-shaped notches 18, 20, which allow the pads 14, 16 to slide along arod 22. Another slide pad 24 attached to the system 10 at the rear(shown in FIG. 3) similarly slides along a rod 26. One or more slidepads 14, 16, and 24 may be replaced by other support elements withdifferent configurations or methods of moving, such as wheels. By theterm “slide pads,” applicants intend to cover any elements allowing thesystem 10 to slide or move back and forth relative to a predeterminedfield-of-view. Slide pads 14, 16 include a height adjustment featureallowing the front corners of system 10 to be raised or lowered forleveling purposes. Specifically, each slide pad 14, 16 is attached tosupport structure 12 by respective threaded rods 28, 30 and nuts 32, 34fixed to the support structure 12. Rods 28, 30 each include a driveportion 28 a, 30 a that may be used to adjust pads 14, 16.

[0035] Referring now to FIGS. 1-3, launch monitoring system 10 furtherincludes first and second camera units 36, 38, a centrally disposedcontrol box 40, and a dual strobe lighting unit 42. First and secondcamera units 36, 38 are preferably ELECTRIM EDC-1000U Computer Camerasfrom Electrim Corporation in Princeton, N.J. Charge coupled device orCCD cameras are preferred but TV-type video cameras are also useful. Theangle between the two cameras' line of sight is preferably in the rangeof 10-30, with 22 being most preferable. Each of the cameras 36, 38 hasa light-receiving aperture, shutter, and light sensitive silicon panel39 (see FIG. 4, showing a silicon panel, which also generallycorresponds to an image captured by the cameras and used by the system).The cameras are directed and focused on a predetermined field-of-view inwhich a golf ball moves and is imaged.

[0036] As shown in a three-dimensional, predetermined, rectilinearfield-of-view (shown in phantom) in FIG. 5, golf ball 41 preferably hassix (6) reflective, spaced-apart round areas or dots 41 a-f placedthereon. Golf ball 41 is shown in two positions I and II to illustratethe preferred embodiment, corresponding to the locations of the golfball 41 when imaged by the system. In positions I and II the golf ballis shown after being struck. The image taken at position I occurs at afirst time and occurs at in position II at a second time. The preferreddiameters of the round dots 41 a-f range from one-tenth ({fraction(1/10)}) to one-eighth (⅛) of an inch, but other sized and shaped areascan be used. Dots 41 a-f are preferably made of reflective materialwhich is adhered to the golf ball. The Scotchlite™ brand beaded materialmade by Minnesota Mining and Manufacturing (3M) is preferred for formingthe dots. Comer-reflective retro-reflectors may also be used.Alternatively, painted spots can be used that define contrasting areas.The number of dots or areas may be as few as three (3) and up to six (6)or more for the golf ball, provided each dot or area reflects light fromthe golf ball in both positions shown in FIG. 5. As a result of thepositioning of the cameras 36, 38 and the dots 41 a-f, both cameras 36and 38 are capable of receiving light reflected by dots 41 a-f, whichappear as bright areas 39 a-f on the silicon panel 39 (as shown in FIG.4) and the corresponding image. Alternatively, the dots may benon-reflective, appearing as dark areas 39 a-f on the silicon panel.

[0037] Reflective materials as compared with the coated surface of thegolf ball can be as high as nine hundred (900) times brighter where thedivergence angle between the beam of light striking the dots 41 a-f andthe beam of light from such dots to the camera aperture is zero or closeto zero. As the divergence angle increases, the ratio of brightness ofsuch dots 41 a-f to the background decreases. It will be appreciatedthat electromagnetic waves outside the range of visible light, such asinfra red light, may be used to make the flash light invisible to thegolfer.

[0038] The control box 40 communicates via an asynchronous protocol viaa computer's parallel port to the camera units 36, 38 to control theiractivation and the dual strobe lighting unit 42 to set off thesuccessive flashes. Dual strobe lighting unit 42 includes two VivitarAutomatic Electronic Flash Model 283 strobe lights mounted on top of oneanother. These strobe lights sequentially direct light onto a beamsplitter 43 and then out of the unit through windows 44 and 46 toreflective elements or panels 48, 50 and then to the predeterminedfield-of-view. Panels 48, 50 may be plates formed of polished metal,such as stainless steel or chrome-plated metal. Other light reflectiveelements may also be used without departing from the spirit or scope ofthe invention. Each reflective panel 48, 50 includes an aperture 52, 54.Cameras 36, 38 are fixed on support structure 56, 58 and are therebydisposed with their respective lenses 60, 62 directed to thepredetermined field-of-view through apertures 52, 54. Video lines 64, 66feed the video signals into control box 40 for subsequent use.

[0039] The locations of the strobe lights, beam splitter, reflectiveelements and cameras allow the light directed from the strobe to enterthe field-of-view and be reflected back from the ball, due to thereflective dots, to the camera lenses through the apertures. In anotherembodiment, ring-shaped strobe lights can be used which surround eachcamera lens. Since the ring-shaped strobe lights are positioned close tothe lenses and the center axis of the strobe is aligned with the centerof the lenses, the light once reflected off the markers would enter thelenses. Thus, eliminating the need for the reflective elements.

[0040] Preferably, telescoping distance calibrators 68, 70 are affixedto support structure 12. The telescoping members are used in calibratinglaunch monitoring system 10 at the appropriate distance from an objectto be monitored. Distance calibrators 68, 70 are extendable members forexample conventional radio antennae can be used. Calibrators 68, 70 areused in conjunction with a calibration fixture shown in FIG. 15 anddiscussed in detail below with respect to the second embodiment. It willbe understood that the same calibration fixture is preferably used withboth the first and second embodiments. At least one distance calibratorshould be used.

[0041] In this first embodiment, a microphone 72 is used to begin theoperation of the system 10. When the golf club hits the golf ball, afirst image of the golf ball 41 in the predetermined field-of-view istaken, as shown in FIG. 5 position I, in response to the sound beingtransmitted by the microphone 72 to the system 10. Since the system 10is preferably used to monitor only the golf ball, although it could alsobe used to monitor the golf club, the first of the two images needs onlyto be taken once the golf ball is struck by the club, as illustrated bythe golf ball in position I of FIG. 5. A laser or other apparatus (notshown) can also be used to initiate the system. For example, theinitiating means can include a light beam and a sensor. When the movinggolf ball passes through the light beam the sensor sends a signal to thesystem. When the laser is used, the laser is arranged such that a golfclub breaks the laser beam just after (or at the time) of contact withthe golf ball. That is, the laser is aligned directly in front of theteed golf ball and the first image taken as or shortly after the golfball leaves the tee. The operation of the first embodiment is discussedin detail below after a description of the second embodiment.

[0042] FIGS. 6-10 illustrate a second embodiment of the presentinvention that further reduces the size and therefore increases theportability of the system.

[0043] Launch monitoring system 100 includes a base or support structure112 that may also have a cover 113. Slide members or pads 114, 116 areutilized at a lower front portion of support structure 112 and includenotches 118, 120 for receiving a rod 190 along which pads 114, 116 mayslide. As shown in FIGS. 7 and 8, wheels 122, 124 replace the pad 24disclosed with respect to the first embodiment shown in FIGS. 1-3.Wheels 122, 124 are attached for rotation and to support structure thatincludes a handle 126 for allowing an operator to move launch monitoringsystem 100 back and forth along the ground. Like the first embodiment,this second embodiment also includes threaded rods 128, 130 andrespective nuts 132, 134 for allowing height adjustment at the front oflaunch monitoring system 100. The wheels may also be height adjustedrelative to the support 112 to allow the system 100 to be adjusteddepending on the terrain on which the system is placed. Although notshown for the second embodiment, the systems in the first and secondembodiments also have a computer and monitor 43 (as shown in FIG. 1).The computer and monitor may be combined into a single element or beseparate elements. The computer has several algorithms and programs usedby the system to make the determinations discussed below.

[0044] As further shown in FIGS. 6 and 7, first and second camera units136, 138 are affixed to support structure 112. These electro-opticalunits 136, 138 are smaller than those disclosed with respect to thefirst embodiment and are preferably the ELECTRIM EDC-1000HR ComputerCameras available from the Electrim Corporation in Princeton, N.J. Thecameras also have light-sensitive silicon panels as in the firstembodiment. The cameras 136, 138 each have a line-of-sight, which areillustrated as solid lines in FIG. 9, that are directed to and focusedon the predetermined field-of-view. As illustrated in FIG. 9 with thebroken lines, the cameras' fields-of-view are larger than are necessaryto image just a single golf ball. Thus, the predetermined field-of-viewis the cameras' fields-of-view at the location where the cameras'lines-of-sight intersect.

[0045] A control box 140 is provided and includes a strobe light unit ata front portion thereof. As shown in FIG. 10, strobe light unit iscomprised of a single flash bulb assembly 144, the related circuitry,and a cylindrical flash tube. The operation of which are described inmore detail below. As best shown in FIG. 6, the reflective elements orpanels 146, 148 are mounted to support structure 112 in a similarorientation to those discussed above with respect to the firstembodiment. Reflective panels 146, 148 also include respective apertures150, 152. Referring to FIGS. 6 and 7, cameras 136, 138 are mounted suchthat the lenses 137, 139 are directed through the respective apertures150, 152 in the reflective panels 146, 148 to the predeterminedfield-of-view. Video lines 154, 156 from the respective electro-opticalunits 136, 138 lead to control box 140. Like the first embodiment, thisembodiment includes distance calibrators also in the form of antenna158, 160, and microphone 162 that also is used to initiate the operationof the system. Again, a laser or other method of initiating the systemcould be used.

[0046] Referring to FIG. 10, the increase in the portability of thesecond system 100 over the first system 10 is also due to the use of asingle flash bulb assembly 144, and associated circuitry in the strobelight unit. The strobe light unit has a single flash bulb assembly 144capable of flashing faster than every 1000 microseconds. The circuitsused with the strobe light unit are the subject of another commonlyassigned application (application Ser. No. 09/008,588), which isincorporated herein in its entirety by express reference thereto. Adiagram of the circuit used for the strobe light unit is illustrated inFIGS. 11A and 11B. As there is only a single flash bulb in the strobelight unit, it will be appreciated that two additional reflectiveelements are required. Referring to FIG. 6, a third light-reflectingpanel 164 reflects about one-half of the light from flash bulb intopanel 146 while a fourth light-reflecting panel 166 reflects the otherhalf of the light into light-reflecting panel 148. The respectiveset-ups for both the calibration mode and the operation mode of system100 are shown in FIGS. 7-8 and 9, respectively.

[0047] To increase the amount of light directed to the reflectiveelements or panels 146, 148, 164, and 166, the system 100 preferably hasan optical or Fresnel lens 168 (as shown in FIG. 12) inserted at thefront of the control box 140, placed between the flash bulb assembly 144and the third and fourth reflective elements or panels 164, 166 as shownin FIGS. 6 and 10. A lens assembly is formed by the lighting unit andthe Fresnel lens. The Fresnel lens 168 directs light from the flash bulbassembly 144 to the third and fourth reflective elements 164, 168. TheFresnel lens has a collimating effect on the light from a cylindricalflash tube. Thus, light pattern with the Fresnel lens 168 controls thedispersion of light as shown in FIG. 14. FIG. 13 shows the light patternwithout the Fresnel lens 168. The lens 168 preferably has a focal lengthof about 3 inches, and the center of the flash bulb assembly 144 is lessthan 3 inches behind the lens. This arrangement allows the system 100 tohave a smaller flash bulb assembly 144 than without the lens 168 becausethe collimation of the light increases the flux of light at the golfball in the predetermined field-of-view. This increase in the fluxallows the possibility of using other reflective materials (or none atall), as well as the use of the system in brighter lighting conditions,including full-sun daylight.

[0048] As shown in FIG. 15, and in use with the system in FIGS. 7 and 8,a calibration fixture 170 is provided to calibrate the system. Althoughthis discussion is with reference to system 100, it applies equally tosystem 10. The fixture 170 includes receiving elements or tabs 172, 174extending outwardly from outer legs 176, 178 for receiving an endportion of the distance calibrators 158, 160. When positioned at thislocation in accordance with the distance calibrators 158, 160, a centralleg 180 of fixture 170 is disposed at the proper location for a golfball 182 used in a launch monitoring operation, as shown in FIG. 9. Golfball 182 also has the pattern of retro-reflective dots as golf ball 41(as shown in FIG. 5) in the first embodiment. Calibration fixture 170further includes an optical level indicator 184 on a top surface thereoffor allowing fixture 170 to be leveled before the calibration procedure.Finally, spikes 186, 188 (as shown in FIG. 8) extending from the bottomof fixture 170 are inserted into the turf to stabilize fixture 170during the calibration procedure. It will be appreciated thatcalibration fixture 170 and golf ball 182 are also preferably used withthe first embodiment shown in FIGS. 1-3 in the same manner discussedhere. In this regard, fixture 170 has a pattern of retro-reflective dots170 a-o, as shown in FIG. 15. Applicants have found that only 15 dots(as opposed to the twenty dots used on the calibration fixture of theparent application-application Ser. No. 08/751,447) are necessary. Sincethe longitudinal movement of the golf ball is greater than its verticalmovement during the time between the two images (see, e.g., FIG. 4), thecalibration of the system need not be as precise in the verticaldirection. Therefore, fewer dots in the vertical direction on thecalibration fixture are needed to adequately calibrate the system.

[0049] As a further means for providing portability to the launchmonitoring systems of the present invention, and as shown in FIGS. 16and 17, rod 190 (which may also be the same as rod 22 for system 10 )may be easily disassembled for transport and reassembled on site beforeoperation of any of the disclosed launch monitoring systems.Specifically, rod 190 may comprise a plurality of sections 190 a-d.Preferably, each of these sections comprises a hollow tube containing asingle elastic cord 192 affixed at opposite ends of rod 190. Cord 192has a relaxed length less than the total length of rod 190 in order tohold sections 190 a-d together. Sections 190 a, 190 b, 190 c haverespective reduced diameter portions 194, 196, 198 that fit withinrespective ends of sections 190 b, 190 c, 190 d. Pins 200, 202 areprovided at opposite ends of rod 190 to allow the rod 190 to be securedinto the turf.

[0050] The use of both systems 10 and 100 is shown generally in FIG. 18.

[0051] At step S101, the system starts and determines if this is thefirst time the system has been used. By default, the system will use thelast calibration when it is first activated. Therefore, the system mustbe calibrated each time the system is moved and/or turned on.

[0052] At step S102, the system is calibrated to define the coordinatesystem to be used by the system.

[0053] After the system is calibrated, the system is set at step S103for either the left- or right-handed orientation, depending on thegolfer to be tested. The selection of the left-handed orientationrequires one set of coordinates are used for the left-handed golfer andright-handed system requires another set of coordinates for aright-handed golfer. At this time, the system is also set up as either atest or a demonstration. If the test mode is selected, the system willsave the test data, while in the demonstration mode it will not save thedata.

[0054] At step S103, additional data specific to the location of thetest and the golfer is entered as well. Specifically, the operatorenters data for ambient conditions such as temperature, humidity, windspeed and direction, elevation, and type of turf to be used in makingthe calculations for the golf ball flight, roll, and total distance. Theoperator also inputs the personal data of the golfer. This personal dataincludes name, age, handicap, gender, golf ball type (for use intrajectory calculations discussed below), and golf club used (type, clubhead, shaft).

[0055] After this data is entered, the system is ready for use and movesto step S104. At step S104, the system waits for a sound trigger fromthe microphone. When there is a sound of a sufficient level or type, thesystem takes two images (as shown in FIG. 4) of the golf ball in thepredetermined field-of-view separated by a short time interval,preferably 800 microseconds, with each of the two cameras 136, 138 (asshown in FIG. 6). The images recorded by the silicon panel 39 are usedby the system to determine the flight characteristics of the golf ball.

[0056] At steps S105-S107, the system uses several algorithms stored inthe computer to determine the location of the golf ball relative to themonitor. After the computer has determined the location of the golf ballfrom the images, the system (and computer algorithms) determine thelaunch conditions. These determinations, which correspond to steps S105,S106, and S107, include locating the bright areas in the images,determining which of those bright areas correspond to the dots on thegolf ball, and, then using this information to determine the location ofthe golf ball from the images, and calculate the launch conditions,respectively. Specifically, the system, at step S105, analyzes theimages recorded by the cameras by locating the bright areas in theimages. A bright area in the image corresponds to light from the flashbulb assembly 144 reflecting off of the retro-reflective dots or markerson the golf ball. Since the golf ball preferably has 6 dots on it, thesystem should find twelve bright areas that represent the dots in theimages from each of the cameras (2 images of the golf ball with 6 dots).The system then determines which of those bright areas correspond to thegolf ball's reflective dots at step S106. As discussed in detail belowwith reference to FIG. 20, this can be done in several ways. If onlytwelve dots are found in the image, the system moves on to step S107 todetermine, from the dots in the images, the position and orientation ofthe golf ball during the first and second images. However, if there aremore or less than twelve dots or bright areas found in the images, thenat step S108 the system allows the operator to manually change theimages. If too few right areas are located, the operator adjusts theimage brightness, and if too many are present, the operator may deleteany additional bright areas. In some instances, the bright areas in theimages may be reflections off of other parts of the golf ball or off thegolf club head. If it is not possible to adequately adjust thebrightness or eliminate those extraneous bright areas, then the systemreturns the operator to step S104 to have the golfer hit another golfball. If the manual editing of the areas is successful, however, thenthe system goes to step S107.

[0057] At step S107, the system uses the identification of the dots instep S106 to determine the location of the centers of each of the twelvedots in each of the two images. Knowing the location of the center ofeach of the dots, the system can calculate the golf ball's spin rate,velocity, and direction.

[0058] At step S109, the system uses this information, as well as theambient conditions and the golf ball information entered at step S103 tocalculate the trajectory of the golf ball during the shot. The systemwill also estimate where the golf ball will land (carry), and even howfar it will roll, giving a total distance for the shot. Because thesystem is calibrated in three dimensions, the system will also be ableto calculate if the golf ball has been sliced or hooked, and how far offline the ball will be.

[0059] This information (i.e., the golfer's launch conditions) is thenpresented to the golfer at step S110, in numerical and/or graphicalformats. At step S111, the system can also calculate the sameinformation if a different golf ball had been used (e.g., a two-piecerather than a three-piece golf ball). It is also possible to determinewhat effect a variation in any of the launch conditions (golf ballspeed, spin rate, and launch angle) would have on the results.

[0060] The golfer also has the option after step S112 to take more shotsby returning the system to step S104. If the player had chosen the testmode at step S103 and several different shots were taken, at step S113the system calculates and presents the average of all data accumulatedduring the test. At step S114, the system presents the golfer with theideal launch conditions for the player's specific capabilities, therebyallowing the player to make changes and maximize distance. The systemallows the golfer to start a new test with a new golf club, for example,at step S115, or to end the session at S116.

[0061] Now turning to the first of these steps in detail (FIG. 19), thecalibration of the system begins with setting up and leveling the systemin step S120. The system is preferably set up on level ground, such as apractice tee or on a level, large field. Obviously, it is also possibleto perform the tests indoors, hitting into a net. Referring to FIGS.6-8, to level the system, the operator uses the threaded rods 128, 130and nuts 132, 134. Referring to FIGS. 7 and 8, the system is positionedto set the best view of the event and the predetermined field-of-view.Then at step S121, the calibration fixture 170 is placed in theappropriate location, which is at the end of the distance calibrators158, 160. The calibration fixture 170 must be level and parallel to thesystem to ensure the best reflection of the light from the flash bulbassembly 144. Placing the calibration fixture at the end of the distancecalibrators 158, 160 ensures that during the test, the calibrationfixture 170 and the golf ball are in full view of each of the cameras.Both cameras take a picture of the calibration fixture and send theimage to a buffer in step S122.

[0062] In step S123, the system, including a calibration algorithm, mustthen determine the location of the centers of the spots in each imagecorresponding to the calibration fixture's retro-reflective dots. In oneembodiment, the system locates the centers of these spots by identifyingthe positions of the pixels in the buffer that have a light intensitygreater than a predetermined threshold value. Since the images aretwo-dimensional, the positions of the pixels have two components (x,y).The system searches the images for bright areas and finds the edges ofeach of the bright areas. The system then provides a rough estimate ofthe centers of each of the bright areas. Then all of the bright pixelsin each of the bright areas are averaged and an accurate dot positionand size are calculated for all 15 areas. Those with areas smaller thana minimum area are ignored.

[0063] Once the location of each of the dots on the calibration fixturewith respect to camera are determined, the system must know the truespacing of the dots on the calibration fixture. As shown in FIG. 15, thecalibration fixture has dots arranged in three rows and five columns.The dots are placed about one inch apart, and on three separate X planesthat are 1.5 inches apart. The X, Y, and Z coordinates of the center ofeach dot 170 a-o, which are arranged in a three-dimensional pattern,were pre-measured to accuracy of one of one-ten thousandth of an inch ona digitizing table and stored in the computer. The system recalls thepreviously stored data of the three-dimensional positions of the dots onthe calibration fixture relative to one another. The recalled datadepends on the whether a right-handed (X-axis points toward the golfer)or a left-handed (X-axis points away from the golfer) system is used.Both sets of data are stored and can be selected by the operator at stepS124. An exemplary set of these three dimensional positions for righthand calibration for the calibration fixture with 15 dots appear below:(1) −1.5 3.0 0.0 (2)  1.5 3.0 1.0 (3)  0.0 3.0 2.0 (4)  1.5 3.0 3.0 (5)−1.5 3.0 4.0 (6) −1.5 2.0 0.0 (7)  1.5 2.0 1.0 (8)  0.0 2.0 2.0 (9)  1.52.0 3.0 (10) −1.5 2.0 4.0 (11) −1.5 1.0 0.0 (12)  1.5 1.0 1.0 (13)  0.01.0 2.0 (14)  1.5 1.0 3.0 (15) −1.5 1.0 4.0

[0064] An exemplary set of these three dimensional positions for lefthand calibration for the calibration fixture with 15 dots appear below:(1)  1.5 3.0 4.0 (2) −1.5 3.0 3.0 (3)  0.0 3.0 2.0 (4) −1.5 3.0 1.0 (5) 1.5 3.0 0.0 (6)  1.5 2.0 4.0 (7) −1.5 2.0 3.0 (8)  0.0 2.0 2.0 (9) −1.52.0 1.0 (10)  1.5 2.0 0.0 (11)  1.5 1.0 4.0 (12) −1.5 1.0 3.0 (13)  0.01.0 2.0 (14) −1.5 1.0 1.0 (15)  1.5 1.0 0.0

[0065] At step S125, using the images of the calibration fixture, thesystem determines eleven (11) constants relating image space coordinatesU and V to the known fifteen X, Y, and Z positions on the calibrationfixture. The equations relating the calibrated X(I), Y(I), Z(I) spacedpoints with the U_(i) ^(j), V_(i) ^(j) image points are: $\begin{matrix}{U_{i}^{j} = \frac{{D_{1j}{X(i)}} + {D_{2j}{Y(i)}} + {D_{3j}{Z(i)}} + D_{4j}}{{D_{9j}{X(i)}} + {D_{10j}{Y(i)}} + {D_{11j}{Z(i)}} + 1}} & \left( {{Eq}.\quad 1} \right)\end{matrix}$

$\begin{matrix}{V_{i}^{j} = \frac{{D_{5j}{X(i)}} + {D_{6j}{Y(i)}} + {D_{7j}{Z(i)}} + D_{8j}}{{D_{9j}{X(i)}} + {D_{10j}{Y(i)}} + {D_{11j}{Z(i)}} + 1}} & \left( {{Eq}.\quad 2} \right)\end{matrix}$

[0066] The eleven constants, D_(i1)(I=1,11), for camera 136 and theeleven constants, D_(i2)(I=1,11), for camera 138 are solved from knowingX(I), Y(I), Z(I) at the 15 locations and the 15 U_(i) ^(j), V_(i) ^(j)coordinates measured in the calibration photo for the two cameras.

[0067] In another embodiment, during image analysis the system uses thestandard Run Length Encoding (RLE) technique to locate the bright areas.The RLE technique is conventional and known by those of ordinary skillin the art. Image analysis can occur during calibration or during anactual shot. Once the bright areas are located using the RLE technique,the system then calculates an aspect ratio of all bright areas in theimage to determine which of the areas are the retro-reflective markers.The technique for determining which bright areas are the dots isdiscussed in detail below with respect to FIG. 20.

[0068] As noted above, once the system is calibrated in step S102, theoperator can enter the ambient conditions, including temperature,humidity, wind, elevation, and turf conditions. Next, the operatorinputs data about the golfer. For example, the operator entersinformation about the golfer, including the golfer's name, the testlocation, gender, age and the golfer's handicap. The operator alsoidentifies the golf ball type and club type, including shaftinformation, for each test.

[0069] A golf ball is then set on a tee where the calibration fixturewas located and the golfer takes a swing. The system is triggered when asound trigger from the club hitting the golf ball is sent via microphoneto the system. The strobe light unit is activated causing a first imageto be recorded by both cameras. There is an intervening, predeterminedtime delay, preferably 800 microseconds, before the strobe light flashesagain. The time delay is limited on one side by the ability to flash thestrobe light and on the other side by the field-of-view. If the timedelay is too long, the field-of-view may not be large enough to capturethe golf ball in the cameras' views for both images. The cameras used inthe systems 10 and 100 allow for both images (which occur during thefirst and the second strobe flashes) to be recorded in one image frame.Because the images are recorded when the strobe light flashes (due toreflections from the retro-reflective material on the golf ball), theflashes can be as close together as needed without concerns for theconstraints of a mechanically shuttered camera.

[0070] This sequence produces an image of the reflections of light offof the retro-reflective dots on each light sensitive panel of thecameras. The location of the dots in each of the images are preferablydetermined with the RLE technique which was discussed for thecalibration fixture.

[0071] The technique used for determining the aspect ratio to determinewhich bright areas are dots will now be described in conjunction withFIG. 20. As shown in step S130, the image must have an appropriatebrightness threshold level chosen. By setting the correct thresholdlevel for the image to a predetermined level, all pixels in the imageare shown either as black or white. Second, at step S131, the images aresegmented into distinct segments, corresponding to the bright areas ineach of the images. The system, at step S132, determines the center ofeach area by first calculating the following summations at each of thesegments using the following equations:

S _(x) =ΣX _(i)  (Eq. 3)

S _(y) =ΣY _(i)  (Eq. 4)

S _(xx) =ΣX _(i) ²  (Eq. 5)

S _(yy) =ΣY _(i) ²  (Eq. 6)

S _(xy) =ΣX _(i) Y _(i)  (Eq. 7)

[0072] Once these sums, which are the sums of the bright areas, havebeen accumulated for each of the segments in the image, the net momentsabout the x and y axes are calculated using the following equations:$\begin{matrix}{I_{x} = {S_{xx} - \frac{S_{x}^{2}}{AREA}}} & \left( {{Eq}.\quad 8} \right) \\{I_{y} = {S_{yy} - \frac{S_{y}^{2}}{AREA}}} & \left( {{Eq}.\quad 9} \right) \\{I_{xy} = {S_{xy} - \frac{S_{x}S_{y}}{AREA}}} & \left( {{Eq}.\quad 10} \right)\end{matrix}$

[0073] where AREA is the number of pixels in each bright area.

[0074] At step S133, the system eliminates those areas of brightness inthe image that have an area outside a predetermined range. Thus, areasthat are too large and too small are eliminated. In the preferredembodiment, the dots on the golf ball are ¼″-⅛″ and the camera has753×244 pixels, so that the dots should have an area of about 105 pixelsin the images. However, glare by specular reflection, including thatfrom the club head and other objects, may cause additional bright areasto appear in each of the images. Thus, if the areas are much less ormuch more than 105 pixels, then the system can ignore the areas sincethey cannot be a marker on the golf ball.

[0075] For those areas that remain (i.e., that are approximately 105pixels) the system determines which are the correct twelve in thefollowing manner. The system assumes that the dots will leave anelliptical shape in the image due to the fact that the dots are roundand the golf ball's movement during the time that the strobe light ison. Therefore, at step S134 the system then calculates the principalmoments of inertia of each area using the following equations:$\begin{matrix}{I_{x^{\prime}} = {\frac{I_{x} + I_{y}}{2} + \sqrt{\left( \frac{I_{x} - I_{y}}{2} \right)^{2} + I_{xy}^{2}}}} & \left( {{Eq}.\quad 11} \right) \\{I_{y^{\prime}} = {\frac{I_{x} + I_{y}}{2} - \sqrt{\left( \frac{I_{x} - I_{y}}{2} \right)^{2} + I_{xy}^{2}}}} & \left( {{Eq}.\quad 12} \right)\end{matrix}$

[0076] Finally, at step S136 the aspect ratio is calculated using thefollowing equation: $\begin{matrix}{R = \frac{I_{x^{\prime}}}{I_{y^{\prime}}}} & \left( {{Eq}.\quad 13} \right)\end{matrix}$

[0077] and the dot is rejected at step S137 if the aspect ratio isgreater than four or five.

[0078] Returning to FIG. 18, once the locations of the dots aredetermined, the system computes the translational velocity of the centerof the golf ball and angular velocity (spin rate) of the golf ball atstep S107 in the following manner. First, the system uses thetriangulation from the data of cameras to locate the position of the sixdots on the surface of the golf ball. Specifically, the system solvesthe set of four linear equations shown below to determine the position(x,y,z) in the golf ball's coordinate system of each dot on the surfaceof the golf ball.

(D _(9,1) U ¹ −D _(1,1))x+(D _(10,1) U ¹ −D _(2,1))y+(D _(11,1) U ¹ −D_(3,1))z+(U ¹ −D _(4,1))=0  (Eq. 14)

(D _(9,1) V ¹ −D _(5,1))x+(D _(10,1) V ¹ −D _(6,1))y+(D _(11,1) V ¹ −D_(7,1))z+(V ¹ −D _(8,1))=0  (Eq. 15)

(D _(9,2) U ² −D _(1,2))x+(D _(10,2) U ² −D _(2,2))y+(D _(11,2) U ² −D_(3,2))z+(U ² −D _(4,2))=0  (Eq. 16)

(D _(9,2) V ² −D _(5,2))x+(D _(10,2) V ² −D _(6,2))y+(D _(11,2) V ² −D_(7,2))z+(V ² −D _(8,2))=0  (Eq. 17)

[0079] where D_(ij) are the eleven constants determined by thecalibration method at steps S102 (FIG. 18) and S125 (FIG. 19), where iidentifies the constant and j identifies the image.

[0080] Next, the system converts the dot locations (determined at stepS135, FIG. 20) in the golf ball coordinate system to the referenceglobal system of the calibrated cameras 136, 138 using the followingmatrix equation: $\begin{matrix}{\left\lfloor \begin{matrix}x_{g} \\y_{g} \\z_{g}\end{matrix} \right\rfloor = {\left\lfloor \begin{matrix}T_{x} \\T_{y} \\T_{z}\end{matrix} \right\rfloor + {\left\lfloor \begin{matrix}M_{11} & M_{12} & M_{13} \\M_{21} & M_{22} & M_{23} \\M_{31} & M_{32} & M_{33}\end{matrix} \right\rfloor \left\lfloor \begin{matrix}x_{b} \\y_{b} \\z_{b}\end{matrix} \right\rfloor}}} & \left( {{Eq}.\quad 18} \right)\end{matrix}$

[0081] where Xg, Yg, Zg are the global coordinates of the center of thegolf ball. The column vector, T_(x),T_(y),T_(z), is the location of thecenter of the golf ball in the global coordinate system. The matrixelements M_(ij)(i=1,3; j=1,3) are the direction cosines defining theorientation of the golf ball coordinate system relative to the globalsystem. The three angles a₁,a₂, a₃ describe the elements of matrixM_(ij) in terms of periodic functions. Substituting matrix equation forthe global position of each reflector into the set of four linearequations shown above, a set of 28 equations result for the six unknownvariables (T_(x),T_(y),T_(z),a₁,a₂,a₃). A similar set of 28 equationsmust be solved for the second image of the golf ball. Typically, thesolution of the three variables T_(x),T_(y)T_(z) and the three angles ata₁,a₂,a₃ that prescribed the rotation matrix M is solvable in fouriterations for the 28 equations that must be simultaneously satisfied.

[0082] The kinematic variables, three components of translationalvelocity and three components of angular velocity in the globalcoordinate system, are calculated from the relative translation of thecenter of mass and relative rotation angles that the golf ball makesbetween its two image positions.

[0083] The velocity components of the center of mass V_(x),V_(y),V_(z)along the three axes of the global coordinate system are given by thefollowing equations:${V_{x} = \frac{{T_{x}\left( {t + {\Delta \quad T}} \right)} - {T_{x}(t)}}{\Delta \quad T}};$${V_{y} = \frac{{T_{y}\left( {t + {\Delta \quad T}} \right)} - {T_{y}(t)}}{\Delta \quad T}};$$V_{z} = \frac{{T_{z}\left( {t + {\Delta \quad T}} \right)} - {T_{z}(t)}}{\Delta \quad T}$

[0084] (Eqs. 19, 20, and 21, respectively) in which t is the time of thefirst strobe measurement of T_(x),T_(y),T_(z) and ΔT is the time betweenimages.

[0085] The spin rate components in the global axis system result fromobtaining the product of the inverse orientation matrix, M^(T)(t) andM(t+ΔT). The resulting relative orientation matrix, A,A(t,t+Δt)=M(t+Δt)M^(T)(t), measures the angular difference of the twostrobe golf ball images.

[0086] The magnitude Θ of the angle of rotation about the spin axisduring the time increment ΔT is given by: $\begin{matrix}{\theta = {\sin^{- 1}\left( \frac{R}{2} \right)}} & \left( {{Eq}.\quad 22} \right)\end{matrix}$

[0087] where

[0088] R={square root}{square root over (l²+m²+n²)};

[0089] l=A₃₂−A₂₃; m=A₁₃−A₃₁; and n=A₂₁−A₁₂.

[0090] The three orthogonal components of spin rate, W_(x),W_(y),W_(z),are given by the following equations: $\begin{matrix}{W_{x} = \frac{\Theta \quad L}{R\quad \Delta \quad t}} & \left( {{Eq}.\quad 23} \right) \\{W_{y} = \frac{\Theta \quad M}{R\quad \Delta \quad t}} & \left( {{Eq}.\quad 24} \right) \\{W_{z} = \frac{\Theta \quad N}{R\quad \Delta \quad t}} & \left( {{Eq}.\quad 25} \right)\end{matrix}$

[0091] At step S109 of FIG. 18, the system, including a computeralgorithm, then computes the trajectories for the tests using theinitial velocity and initial spin rate which were computed in step S107.For each time increment, the system interpolates the forces on the golfball at time T and calculates the velocity at time T+1 from the velocityof the golf ball and the forces on the golf ball at time T. Next, thesystem computes the mean velocity and the Reynold's number, which is theratio of the flow's inertial forces to the flow's viscous forces duringthe time interval from time T to time T+1. The system then interpolatesthe mean forces, from which the system calculates the velocity at timeT+1. The forces include the drag force, the lift due to the spin of thegolf ball, and gravitational forces. Using the velocity at time T+1, thesystem can compute the position at time T+1. Finally, the systemcomputes the spin rate at time T+1. In the preferred embodiment, thelength of the time interval is 0.1 seconds. This calculation isperformed until the golf ball reaches the ground.

[0092] The system uses the following equations to perform thesecalculations. For the drag force on the golf ball, the force iscalculated by:

F _(d) =c _(d)*1/2*ρ*|V ^(Bf)|^(2*) A;  (Eq. 26)

[0093] where

[0094] c_(d)=drag coefficient previously determined and stored in a datafile that is called when the golf ball type is selected;

[0095] ρ=density of air—entered at step S103, the beginning of the test;

[0096] |V^(Bf)|=magnitude of the velocity of the golf ball; and

[0097] A=the cross-sectional area of the golf ball—also known from thegolf ball selected.

[0098] The lift, caused by the spin of the golf ball, is perpendicularto the velocity direction and spin direction and is given by:

n _(L) =N _(ω) x n _(VB),  (Eq. 27)

[0099] where n_(L), N_(ω), and n_(VB) are the direction cosines of thelift force, the angular rotation of the golf ball, and the velocity ofthe golf ball, respectively.

[0100] The magnitude of the lift is given by:

F _(L) =c _(L)*1/2*ρ*|V ^(Bf)|^(2*) A  (Eq. 28)

[0101] where c_(L) is the lift coefficient and the other terms beingdefined above.

[0102] Therefore, the applied aerodynamic force on the golf ball becomes

R ^(B) =n _(L) F _(L) −n _(VB) F _(d)  (Eq. 29)

[0103] The velocity and spin of the golf ball are then transformed intothe X, Y, and Z directions so that generalized velocities and rotationalvelocities are given by

V ^(Bf) =u ₉ X+u ₁₀ Y+u ₁₁ Z  (Eq. 30)

ω ^(Bf) =u ₁₂ X+u ₁₃ Y+u ₁₄ Z  (Eq. 31)

[0104] where u₉, u₁₀, and u₁₁ are the velocities in the X, Y, and Zdirections; and u₁₂, U₁₃, and U₁₄ are the spin velocities in the X, Y,and Z directions.

[0105] Using these equations, the system obtains the following secondorder differential equations:

n _(1x) *F ₁ −n _(Vbx) *F _(d) −m _(B) *u ₉=0  (Eq. 32)

n _(1y) *F ₁ −n _(Vby) *F _(d) −m _(B) *u ₁₀=0  (Eq. 33)

n _(1z) *F ₁ −n _(Vbz) *F _(d) −m _(B) *u ₁₁ −m _(B) *g=0  (Eq. 34)

[0106] where

[0107] n_(1x), n_(1y), n_(1z) are the direction cosines of the force inthe X, Y, and Z directions, respectively;

[0108] n_(Vbx), n_(Vby), and n_(Vbz) are the directions of the velocityvectors in the X, Y, and Z directions, respectively;

[0109] m_(B) is the mass of the ball; and

[0110] m_(B)*g relates to the gravitational force exerted on the golfball in the Z direction.

[0111] These second order differential equations are then solved foreach time step, preferably every 0.1 second using the drag and liftcoefficients (C_(d) and C_(L)) from data files, or from another source,based upon the velocity (V^(Bf)) and angular velocity (ω^(Bf)) at eachof those time steps.

[0112] The trajectory method repeats this procedure for successive timeintervals until the computed elevation component of the golf ball'sposition is less than a predetermined elevation, usually zero or groundlevel. See FIG. 21. When the golf ball reaches ground level, the methodinterpolates to compute the ground impact conditions including finalvelocity, trajectory time, impact angle, and spin rate. Using a rollmodel based on empirical data and golf ball data input by the operator,the system computes the final resting position of the golf ball usingthe just-computed ground impact conditions. Accordingly, the systemcomputes the total distance from the tee to the final resting positionof the golf ball. A data file stores the results computed by thetrajectory method.

[0113] The system then determines whether an additional test will beperformed. If additional tests are to be performed, the processdescribed above repeats, beginning at step S104 with the sound triggerthrough step S110 where the trajectory method computes and resents thetrajectory for the golf ball.

[0114] When all tests have been performed, the analysis method computesstatistics for each golf ball type used in the tests and presents theresults to the operator. For the group of tests performed for each golfball type, the system computes the average value and standard deviationfrom the mean for several launch characteristics including the velocity,the launch angle, the side angles, the backspin, the side spin, and thecarry and roll.

[0115] Different factors contribute to the standard deviation of themeasurements including the variation in the compression and resilienceof the golf balls, the variation in the positioning of the dots on thegolf balls, the pixel resolution of the light sensitive panels and theaccuracy of the pre-measured dots on the calibration fixture. Obviously,the primary source of scatter lies in the swing variations of thetypical golfer.

[0116] Upon request from the operator, the system will display the testresults in various forms. For example, the system will displayindividual results for the golf ball type selected by the operator.

[0117] Similarly, the system in step S113 can also display tabularrepresentations of the trajectories for the golf ball types selected bythe operator. The tabular representation presents trajectory informationincluding distance, height, velocity, spin, lift, drag, and theReynold's number. Similarly, the analysis method displays graphicalrepresentation of the trajectories for the golf ball types selected bythe operator. The system computes the graphical trajectories from theaverage launch conditions computed for each golf ball type.

[0118] At step S113, the system displays the average of each of theshots taken by the golfer. The results are displayed in a tabular and/orgraphical format. The displayed results include the total distance, thespin rate, the launch angle, distance in the air, and golf ball speed.From this information, the system at step S114 shows the golfer theresults if the launch angle and spin rate of the golf ball were slightlychanged, allowing the golfer to optimize the equipment and/or swing. Asshown in FIG. 22, the distance the golf ball travels is dependent on theinitial spin rate and launch angle for a given golf ball speed. It isassumed that the golfer will not be able to increase the golf ballspeed, which in turn is determined by the club head speed. (One way toincrease the golf ball and club head speed, is to increase the shaftlength. However, increasing the shaft length may change other variables,including the launch angle and spin rate, so a new set of tests shouldbe done.)

[0119] At step S114, the system calculates the distances of a golf ballstruck at a variety of launch angles and spin rates that are close tothose for the golfer. The operator is able to choose which launch anglesand spin rates are used to calculate the distances. One example is shownin FIG. 22. In this example, the system calculated the distances that agolf ball having an initial velocity of 130 mph will travel for launchangles of 0° to 15° and having initial spin rates from 2000 rpm to 4000rpm. In order to display this particular data, the system performs thetrajectory calculations described above between about 50-100 times(several predetermined values of launch angles and several predeterminedvalues of initial spin rates). The operator can dictate the range oflaunch angles and spin rates the system should use, as well as how manyvalues of each the system uses in the calculations. From the graphicaldata in FIG. 22, the golfer can determine which of these two variablescould be changed to improve the distance.

[0120] Using FIG. 22, if the golfer had a launch angle of L₂ degrees anda spin rate of S₂ rpm, the golfer would attain a distance within therange of 220 to 225 yards. If the golfer reduced the spin rate from S₂rpm to S₁ rpm, the distance attained would increase to the range of 225to 230 yards. Similarly at a spin rate of S₁ a change in the launchangle from L₁ degrees to L₂ degrees would also increase the distance.The spin rate and launch angle can be altered simultaneously to changethe distances. Knowing this information, the golfer can make theappropriate adjustments to achieve this increase in distance.

[0121] Since the golfer's data is saved, when the system is in the testmode, it is also possible to compare the golfer's data with that ofother golfers, whose data were also saved. In this way, it is possiblefor golfers to have their data (launch angle, initial golf ball speed,spin rate, etc.) compared to others. This comparison may be done in atabular or graphical format. Similarly, the system may compare the datafrom successive clubs (e.g., a 5-iron to a 6-iron to a 7-iron) todetermine if there are gaps in the clubs (inconsistent distances betweeneach of the clubs). Alternatively, two different golfers could becompared using the same or different clubs, or the same or differentballs.

EXAMPLE

[0122] After calibration, a golf machine struck six balata wound golfballs and six two-piece solid golf balls under the same conditions. Thefollowing data for golf ball movement was obtained: Launch W_(x) W_(y)W_(z) Ball Speed Angle Side Angle Rate Rate Rate Units mph degreesdegrees rpm rpm rpm Average 156.7 8.5 −0.7 −4403  3 193 (Wound) Standard0.8 0.4 0.2  184  78 115 Deviation Average 156.6 8.8 −0.7 −3202  3 −23(Two-Piece) Standard 1.0 0.3 0.2  126 197 137 Deviation

[0123] These results illustrate the effect of two different golf ballconstructions on launch conditions. The launch variable primarilyaffected is the resulting backspin of the golf ball (W_(x) rate) onsquarely hit golf shots. A secondary effect is the lower launch angle ofwound construction versus two-piece solid golf balls with high modulusionomer cover material.

[0124] While the above invention has been described with reference tocertain preferred embodiments, it should be kept in mind that the scopeof the present invention is not limited to these embodiments. Oneskilled in the art may find variations of these preferred embodimentswhich, nevertheless, fall within the spirit of the present invention,whose scope is defined by the claims set forth below.

1. A method of processing images of a golf ball having markerscomprising the steps of: capturing at least one bright image of the golfball; dividing the image into a plurality of pixels; assigning a colorto each pixel, the assigned color being either a first color if thepixel has a brightness above a predetermined threshold level or anothercolor if the pixel has a brightness below the predetermined thresholdlevel; connecting adjacent pixels of the first color to form brightareas; corresponding the pixels to identify the markers on the object;and calculating the location of the markers from the at least one image.2. The method of claim 1, wherein the capturing step further comprisesthe steps of: capturing a first pair of images on a first image frame ofa first camera; and capturing a second pair of images on a second imageframe of a second camera.
 3. The method of claim 2, wherein thecalculating step further comprises the step of mathematically comparingthe first pair of images with the second pair of images.
 4. The methodof claim 1, wherein the corresponding step further comprises the step ofeliminating extraneous pixels from the image.
 5. The method of claim 1,further comprising determining at least one of the transverse velocityand spin rate of the golf ball.
 6. The method of claim 1, wherein thedividing includes dividing the image into rows and columns of pixels. 7.The method of claim 6, wherein the dividing further includes dividingthe image into 753 pixels by 244 pixels.
 8. The method of claim 1,further comprising: determining if any of the bright areas are aberrantbright areas; and ignoring the aberrant bright areas.
 9. The method ofclaim 8, wherein the determining includes determining whether any of thebright areas have an area outside a predetermined range of areas. 10.The method of claim 1, further comprising the step of calculating thecenters of the markers.
 11. The method of claim 10, wherein the step ofcalculating the centers of the markers comprises performing a pluralityof summations for the areas of brightness and calculating net momentsabout two axes.
 12. The method of claim 1, wherein about 105 pixelscorrespond to one marker.
 13. A method of processing images of a golfball having at least one marker comprising the steps of: capturing atleast one image of the golf ball; dividing the image into a plurality ofpixels; corresponding an area of the pixels with the marker on the golfball; calculating the location of the marker on the golf ball from theat least one image.
 14. The method of claim 13, further comprising thestep of calculating the center of the area corresponding to the markeron the golf ball.
 15. The method of claim 14, wherein the step ofcalculating the center of the area corresponding to the marker on thegolf ball comprises performing a plurality of summations for the areaand calculating net moments about two axes.
 16. The method of claim 13,further comprising the step of excluding extraneous pixels from the areacorresponding to the marker on the golf ball.
 17. The method of claim13, further comprising determining the principle axis of the areacorresponding to the marker on the ball; calculating a ratio of theprinciple axis of the area corresponding to the marker on the ball; andeliminating the area of pixels if the ratio of the principle axisexceeds a predetermined value.